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Theorem sotri2 6135
Description: A transitivity relation. (Read 𝐴𝐵 and 𝐵 < 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
sotri2 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)

Proof of Theorem sotri2
StepHypRef Expression
1 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
21brel 5743 . . . 4 (𝐵𝑅𝐶 → (𝐵𝑆𝐶𝑆))
32simpld 494 . . 3 (𝐵𝑅𝐶𝐵𝑆)
4 soi.1 . . . . . . 7 𝑅 Or 𝑆
5 sotric 5618 . . . . . . 7 ((𝑅 Or 𝑆 ∧ (𝐵𝑆𝐴𝑆)) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
64, 5mpan 689 . . . . . 6 ((𝐵𝑆𝐴𝑆) → (𝐵𝑅𝐴 ↔ ¬ (𝐵 = 𝐴𝐴𝑅𝐵)))
76con2bid 354 . . . . 5 ((𝐵𝑆𝐴𝑆) → ((𝐵 = 𝐴𝐴𝑅𝐵) ↔ ¬ 𝐵𝑅𝐴))
8 breq1 5151 . . . . . . 7 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
98biimpd 228 . . . . . 6 (𝐵 = 𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶))
104, 1sotri 6133 . . . . . . 7 ((𝐴𝑅𝐵𝐵𝑅𝐶) → 𝐴𝑅𝐶)
1110ex 412 . . . . . 6 (𝐴𝑅𝐵 → (𝐵𝑅𝐶𝐴𝑅𝐶))
129, 11jaoi 856 . . . . 5 ((𝐵 = 𝐴𝐴𝑅𝐵) → (𝐵𝑅𝐶𝐴𝑅𝐶))
137, 12syl6bir 254 . . . 4 ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴 → (𝐵𝑅𝐶𝐴𝑅𝐶)))
1413com3r 87 . . 3 (𝐵𝑅𝐶 → ((𝐵𝑆𝐴𝑆) → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
153, 14mpand 694 . 2 (𝐵𝑅𝐶 → (𝐴𝑆 → (¬ 𝐵𝑅𝐴𝐴𝑅𝐶)))
16153imp231 1111 1 ((𝐴𝑆 ∧ ¬ 𝐵𝑅𝐴𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wss 3947   class class class wbr 5148   Or wor 5589   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-po 5590  df-so 5591  df-xp 5684
This theorem is referenced by:  supsrlem  11135
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